Is there a error?

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Let $G$ be a finite nilpotent group.



We know that $G=G_p_1times G_p_2times cdots times G_p_r$ where $G_p_iin Syl_p_i(G)$, $i=1,dots,r$.



Is the following equation right? And why?



$G/Phi(G)=G_p_1/Phi(G_p_1)timescdotstimes G_p_r/Phi(G_p_r)$,



where $Phi(G)$ is the Frattini subgroup of $G$.




finite-groups




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  • 1




    What's $Phi(G)$? Frattini subgroup?
    – Lord Shark the Unknown
    Jul 26 at 9:42










  • @Lord Shark the Unknown Yes!
    – Qin
    Jul 26 at 9:43






  • 1




    The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
    – Lord Shark the Unknown
    Jul 26 at 9:49







  • 2




    Yes: An "n" is missing.
    – Christian Blatter
    Jul 26 at 10:08














up vote
0
down vote

favorite












Let $G$ be a finite nilpotent group.



We know that $G=G_p_1times G_p_2times cdots times G_p_r$ where $G_p_iin Syl_p_i(G)$, $i=1,dots,r$.



Is the following equation right? And why?



$G/Phi(G)=G_p_1/Phi(G_p_1)timescdotstimes G_p_r/Phi(G_p_r)$,



where $Phi(G)$ is the Frattini subgroup of $G$.




finite-groups




share|cite|improve this question

















  • 1




    What's $Phi(G)$? Frattini subgroup?
    – Lord Shark the Unknown
    Jul 26 at 9:42










  • @Lord Shark the Unknown Yes!
    – Qin
    Jul 26 at 9:43






  • 1




    The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
    – Lord Shark the Unknown
    Jul 26 at 9:49







  • 2




    Yes: An "n" is missing.
    – Christian Blatter
    Jul 26 at 10:08












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $G$ be a finite nilpotent group.



We know that $G=G_p_1times G_p_2times cdots times G_p_r$ where $G_p_iin Syl_p_i(G)$, $i=1,dots,r$.



Is the following equation right? And why?



$G/Phi(G)=G_p_1/Phi(G_p_1)timescdotstimes G_p_r/Phi(G_p_r)$,



where $Phi(G)$ is the Frattini subgroup of $G$.




finite-groups




share|cite|improve this question













Let $G$ be a finite nilpotent group.



We know that $G=G_p_1times G_p_2times cdots times G_p_r$ where $G_p_iin Syl_p_i(G)$, $i=1,dots,r$.



Is the following equation right? And why?



$G/Phi(G)=G_p_1/Phi(G_p_1)timescdotstimes G_p_r/Phi(G_p_r)$,



where $Phi(G)$ is the Frattini subgroup of $G$.





finite-groups





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 26 at 9:44
























asked Jul 26 at 9:38









Qin

8217




8217







  • 1




    What's $Phi(G)$? Frattini subgroup?
    – Lord Shark the Unknown
    Jul 26 at 9:42










  • @Lord Shark the Unknown Yes!
    – Qin
    Jul 26 at 9:43






  • 1




    The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
    – Lord Shark the Unknown
    Jul 26 at 9:49







  • 2




    Yes: An "n" is missing.
    – Christian Blatter
    Jul 26 at 10:08












  • 1




    What's $Phi(G)$? Frattini subgroup?
    – Lord Shark the Unknown
    Jul 26 at 9:42










  • @Lord Shark the Unknown Yes!
    – Qin
    Jul 26 at 9:43






  • 1




    The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
    – Lord Shark the Unknown
    Jul 26 at 9:49







  • 2




    Yes: An "n" is missing.
    – Christian Blatter
    Jul 26 at 10:08







1




1




What's $Phi(G)$? Frattini subgroup?
– Lord Shark the Unknown
Jul 26 at 9:42




What's $Phi(G)$? Frattini subgroup?
– Lord Shark the Unknown
Jul 26 at 9:42












@Lord Shark the Unknown Yes!
– Qin
Jul 26 at 9:43




@Lord Shark the Unknown Yes!
– Qin
Jul 26 at 9:43




1




1




The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
– Lord Shark the Unknown
Jul 26 at 9:49





The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
– Lord Shark the Unknown
Jul 26 at 9:49





2




2




Yes: An "n" is missing.
– Christian Blatter
Jul 26 at 10:08




Yes: An "n" is missing.
– Christian Blatter
Jul 26 at 10:08















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